Method of making speed-reduction gearing



1956 o. E. SAARI 2,731,83fi

METHOD OF MAKING SPEED-REDUCTION GEARING Original Filed July 12, 1954 6 Sheets-Sheet l 0. E. SAARI METHOD OF MAKING SPEED-REDUCTION GEARING Jan. 24, 1956 Original Filed July 12, 1954 6 Sheets-Sheet 2 Jan. 24, 1956 o. E. SAARI METHOD OF MAKING SPEED-REDUCTION GEARING Original Filed July 12, 1954 6 Sheets-Sheet 3 WM r w 2 CM 0% A \J wk N hm MN Jan. 24, 1956 METHOD OF MAKING SPEED-REDUCTION GEARING OIE. SAARI 2,731,886

Jan. 24, 1956 o. E. SAARI 2,731,886

METHOD OF MAKING SPEED-REDUCTION GEARING Original Filed July 12, 1954 6 Sheets-Sheet 5 J5 INVENTOR.

Jan. 24, 1956 O SAAR] 2,731,886

METHOD OF MAKING SPEED-REDUCTION GEARING Original Filed July 1?, 1954 6 Sheets-Sheet 6 R550; 779 N7 DRIVING F0)? 05 IN V EN TOR.

United States Patent Office 2,731,886 METHOD OF MAKING SPEED-REDUCTION GEARENG Gliver E. 'Saari, Chicago, .IiL, assignor to 'Iilinois Tool Wrks,'Chicago, 11]., a corporation of lliinois 2 Claims. (CL 90-4) Serial No. 442,553, filed in the United States Patent Office on July 12, 1954, now Patent No. 2,696,125.

The commonly used type of reduction gearing in which a worm engages teeth on the periphery of a gear requires very accurate positioning of the axes of the'gear and the worm and'cannot easily be adjusted to regulate backlash after wear. The area of tooth engagement is small so that the gearing must be made large in order to transmit a substantial amount of power.

Gearing in which a worm engages teeth on the face of a crown or bevel gear has been proposed, but, in order cred-necessary to give the worm some peculiar form which makes it difficult and expensive to produce. Thus, in some instances the worm has had an excessively large taper, while in others the surface or the thread of the worm has hadpeculiarities which make it expensive to produce, and in most such gearing a high reduction ratio has not been obtained.

The reduction gearing which 'I have invented and which may be produced by practicing my new method consists engagethe thread of the worm throughout substantially The gearing, therefore, has a large power transmitting capacity in comparison with its size.

Heretofore it has not been thought possible to obtain so that by giving the lead a critical value hereinafter defined the tooth engagement is made nearly complete.

Batented Jan. 24, 1956 In order to make the nature of my invention plain, I will describe a specific example of a reduction gearing capable of being produced by employing the -method of my present invention. In this description I shall refer to the accompanyingdrawings in which:

Figs. 1 and '2 are diagrams of gearing embodying the important dimensions by whichthe determined. Fig. 1 is drawnon .a plane which contains-the axisof theworm and the common perpendicular to the worm and gear axis. Fig. .2 is taken on the plane parallel to the worm and gear taxes and perpendicular to the common perpendicular to the axes;

Fig. 3 is a face teeth of the gear of the tooth;

Fig. 3A is a transverse section of taken on the line view of the concaveside of one ofthe looking from the center of curvature either direction;

Fig. 6 is a side elevation of the-gearing embodyingthe invention in which the angle between the axes of .the worm and gear is and shows a mounting for the worm and gear;

Fig. 7 is a vertical section on the line 7-7 of Fig. 8 is a perspective of a portion of the gear axial section of the worm;

Figs. -9 and :10 are diagrammatic .front and end views respectively, of apair of mating skew axis pitch surfaces;

Figs. 1:1 and :12 are also diagrammatic front and end views, respectively, graphically illustrating the development of tooth curves on skew pitch surfaces;

Fig. 6; with an Fig. 14'is a developed view of the conical surface of'the worm.

The gearing illustrated consists of teeth 11 'on one side conical surface containing the bottomsurface of its'thread 21 at the opposite side of this line. the axes of the worm and gear are perpendicular, the common perpendicular to the axes lies in a plane which is coincident with the gear axisand extends .normal :to the worm axis, and in this case the .worm is located on one side of this plane and the apices 28, 29 on the other side of this plane.

The following dimensions indicated in Figs. 1 and '2 may be used to specifyaccuratelytheposition of the worm with respect to the gear:

The distance C between the axis of the worm and the axis of the gear measured .along the common perpendicular to the two axes, which is less than R, the outer radius of the gear;

The angle E between the aXes of the worm and gear measured from the worm axis in a plane parallel to both axes asshown in Fig. 2. This angle should be between 45 and .135", and in many cases may most conveniently be made 90;

The distance x1 from the smaller end of the worm to the point of the common perpendicular 6; V

The distance x2 from the larger end of the worm to the point 0 of the common perpendicular 6;

f The distance a from the point 0 of the common perpendicular 6 to the apex of the conical surface containing the outer surface of the thread of the worm. r

The form of the worm 20 is the frustum of a cone of moderate, as. distinguished from abrupt, taper. The 'taper angle of the worm, that is,.the angle T between its side and its axis, is small. 'The thread 21 of the worm 20 is of uniform lead, L, and uniform cross-sectional shape and height. For multiple-start worms, the axial pitch, or distance between adjacent threads, is equal to the lead divided by the number of threads, or

as shown on the drawing. In the particular worm illustrated, the thread 21 is a triple thread so that the axial pitch is The cross-sectional shape of the thread is determined by the pressure angle h on the high side of the thread, that is, the side facing the larger end of the worm, and the pressure angle 1 on the low side of the thread, that is, the side facing the smaller end of the worm.

The gearing may easily be made by using ordinary cutting and grinding machinery to form the frusto-conical wow 20 and a hob of the same size and shape as the worm. This hob is then used to cut the teeth 11 of the gear by feeding it into a positional relationship to the gear blank which is the same ship of the worm to the gear in the completed gearing illustrated in Figs. 1 and 2; During hobbing, the hob and the gear blank are rotated on their axes. The number of revolutions of the hob during each revolution of the gear blank determines the ratio between the number of teeth formed on the gear blank and the number of threads of the hob and thus determines the speed reduction ratio K which the finished gearing will have.

The hobbing operation which has been described will in general result in forming teeth on the gear having objectionableinactive fillets or undercut portions so that a satisfactory gear will not be produced. I have discovered that these objectionable features may be avoided by giving the lead of the thread of the worm and the hob a critical relationship to the gear reduction ratio and the coordinates determining the position and shape of the worm. In this way, useful cooperation between the thread of the worm may be extended over substantially the entire side surfaces of the gear teeth.

The critical value of the lead L of the thread of the worm measured along its axis is given by the following equation:

' L 21r(C-P sin K csc E-E'; cos 5+ sin -1) cot E] where P=(a+x) tan T and B-AJW as the positional relationwhere A -c0t E cot T and so that it represents the distance from the common perpendicular to a point of the worm between its middle and its smaller end. The lead must also have the proper hand. For the relative position of the Worm and gear shown in the drawings, the lead must be right hand. If the gear is made to bear on the other side of the worm as shown in dotted lines in Fig. 2, or any other change is made in the mounting which results in a similar change of relative position, the lead must be left hand.

, The angle between the axis of the gear and the axis of the worm is usually determined by the arrangement which is most convenient in the machinery in which the reduction gearing is to be used. In many cases, it is convenient to place these axes at right angles. In this case, the formula for the critical lead may be simplified. When the angle E between the axes is cot E becomes 0 and csc E becomes 1. Equation 1 then reduces to In using Equation 2, as in using Equation 1, x may be given any value between x1 and am, but is preferably given a value between an and the pitch point is located on the inner half of the worm and the inner half of the face of the gear.

In order that the critical lead specified in the above equations may have its intended effect of substantially avoiding inactive fillets in the teeth of the gear, the speed 7 reduction ratio should be greater than 9:1, and some care should be exercised in selecting the position and shape of the worm.

The gear ratio K of the gearing illustrated is 20. The gear has sixty teeth and the worm three threads or a triple thread. As the gear ratio is reduced the worm becomes more difiicult to manufacture and the hob, even if of the critical lead, tends to create fillets on the gear teeth. For these reasons, the gear ratio should not be made less than 9. On the other hand, the gear ratio may be increased above 20 without introducing any difficulties or disadvantages until it is made so great that the thread and teeth become too fine to transmit power adequately. Gear ratiosas high as 200 are practicable on large gears.

The taper angle T of the cone in the gearing illustrated is 10. It is'desirable to make the taper angle as large as 10 or even as great as 15 when gear ratios are under 15. On the other hand, in the case of large gear ratios the 1 angle is'more testab e-r h s t whish wi h. t iea tfin Nae- 1 e l M th o the worm' 'may be givenany value d'esired provided, of course,

that the length of the worm; x2 'x1 must be great enough to dacross the teeth of the face of thegear In the specific example illustrated, x1=2 inches; xz= inches, and q=4inches. v 1v V V Ihe pressure angle 9511 on the high side of the thread of' the worm illustrated is 30", This value'is not critical, but to avoid inactive fillets on the gear. the high side pressu'l're angle" shouldfbe at leastZO". The angle should be held below 45 in order to avoid unnecessarily high tooth pressures in the operation of the gearing. I I v The pressure angle on the low side 1 of the thread in thegearin g illustrated is This value is not critical the angle may be reduced to Q although this tends to introduce difiiculties in grinding the'worm. On the other hand,'the angle should be kept below in order to avoid short tooth profiles andhigh tooth pressures.

4 e ratio C/R of the distance between. the axes to the outer radius of the gear determines b oth the extent of the tooth contact zone and the tooth pressures required. In

the gearing illustrated, the outer radius R of the gearis 6% inches and the distance C is 4 inches so that the ratio C/R 150543, but it is not critical. C should always be less than R but, it need be only slightly less than R when the axis angle E is small and may be only one-third of ll when E is large. When the axes are at right angles, the

most advantageous value of C /R depends upon the gear :3

ratio and it may besaid in general that when the gear ratio above that of the gearing illustrated, it is desirable to make'the ratio C/R between 0,8 and 0.6, whilefor lower gear ratios this ratio C/R is most desirably' between 0.6

and 0.4. This places the pitch point on a"ra'dius of t he gear at an angle from to to a plane coincident with :heger'axi and Pa e to h q m a is-t .e

n Tillie importance of thecritical value for the lead ofthe' worm in gearing having a gear ratio and dimensions with in'the limits abovespecified is illustrated in Figs. 3, 4 and 5 Figs. 3', 4 and 5 show one of the side faces of aftooith cut on'the gear by the thread of the hob. Figs. 3A and 4A show the cr ossrsectional shape of the tooth. The faces of the ihac tive fillets 11x, 11y formed on the tooth are indicated by the dark areas in the face views. Fig. 3 shows the form of the teeth out on the gear in the gearing which 1s illustrated. Figs. 4 and 5 show thefoim of teeth which would be cut on the gear in a gearing of precisely the same dimensions if the lead departed from the critical value. Figs.v4 and 4 A show the efi'ect of increasing the v lead above the critical yalue. J Fig. 5 shows messed of maki r g the lead below the critical value .It s'houl d bie noticed that the teeth shown in Figs. 4 and j have inoper ativc fillets which materially reduce their operative faces.

Fig. 3, on the other hand, shows that when'jhe critical value of the lead is used, substantially the entire side surfaces of the gear teeth are operative to contact with the thread of the worm and no' appreciable fillets are formed. This complete tooth contact gives the gearing a high power transferring ability. Also, the absence of undercuts and fillets on the gear makes iteasy to mold a matrix on the gear from which plastic gears may be cast or pressed. The reduction gearing which has been described has physical diflferences which distinguish it from the gearings new in cbmmon" use as well as various" types whichhave been proposed and possess important practical advantages over such'gearing: l 1 I I v l A lhe areaiof contact between the thread of the worm and tlilteeth of the gear is large both because the side faces of the teeth engage the thread throughout substantiallytheir entire areas and because anumbegor helices ofthe worm thread are engaged simultaneously by teeth at e g a T e d 11 Worm 1 .59 mal a'tt v i has rnore one complete thread helix, and may easily be given a number of complete thread helices by 6 u ng a multiple st artthread such as the triple thread ofcontact 3 1 betweerr the thread and the gear teeth are transverse a'nd nearly perpefi'dlcu 1" the direction of the relative sliding movement of nreeflf (as ifidicated in F 8) so that the oil tiln'l carri is used. This is not of or naryiw orm Whl lelt is true m-someextfidf gearing such gearing segn' zgag 7 l l hypbid e'fof tli'lar ge speed se i ,-v r r .nlhthvfsafiglffh worm and the' eomplemen'tary teeth 1:1 of the gear is diclo'sed It will noted th'at the conveit surface of the wonn thread and" the complementary comes; serrate of the gear tooth enga'g e to impart driving force tot the gear. There is nor-trial force as indicated in Fig 13",

tion,' bu t is also capable of imparting reverse rotations to li Worm angig'far' is a rect relatioii their are 7 plerpen r. The: mountin ihclude si a sharpie 6 vih'ichihe gear 10' is d ured i nm'gk' at in a frame 30 and containing roller bearings 14 for the geai' 0 to wh ch are separate gear and The gear assembly in des "eat one we a a as he w me a. t qrr 'lfl sn u awe-nabs 23 s cured in an opening in one elicl of the frame 3i)".

.1. r g .s' i l tr ndlfi 9f t s W r t is it 2.19%? th 'le s id against the inner races of the ball uniform cross-section h le the body 611 which if is formed has a' gradual tap er B'au's'e of this formation the successive turns o f the thread oi the worm differ fro? one another only slight increment in radius This iheahs tha't when the isimouhted acrfossthe fa the gear, thedepth to which the turns of the thread ente n the teeth be vafid by merit of the and that; at}

5 airis'the direct contact of the teeth owing to the identity in form of successive turns of the worm. This results in three special advantages:

, (1) In the first place the very exact setting of the gear and worm axes required in ordinary worm gearing is unnecessary for after the shafts and bearings have been assembled, the penetration of the worm thread between the gear teeth may be adjusted by merely adjusting the axialposition of the worm. It is because of this that the worm and gear with their shafts and bearings may form separate assemblieswhich may be separately secured to a supporting frame as shown in Figs. 6 and 7.

(2) A second advantage is that a similar axial adjustment of the worm may be used to regulate backlash and particularly to take up backlash after wear. Thus, with the mounting shown in Fig. 6, backlash resulting from wear may be taken out by merely loosening the nut and inserting a shim or spacer between the large end 27 of the worm and the inner races of the ball bearings 24.

(3) A third advantage efiects an economy of manufacture for it arises. during'the cutting of the teeth on the gear. In the case of ordinary worm gearing the sharpening of the thread of the hob which cuts the gear necessarily changes the shape and depth of the teeth formed on the gear so that it is impossible to use a single hob to cut a considerable number of gears accurately. This disadvantage is avoided in making the present gearing, for after the hob for cutting the gear teeth has been sharpened it is necessary only to make a slight axial adjustment of the hob to enable it to cut teeth of precisely the same form and the same depth as those which it produced before it was sharpened.

A further advantage arises from the fact that the distance between the axes of the gear and worm is much less than the radius of the gear instead of being greater than the gear radius as in the case of ordinary worm reduction gearing. It is the relatively short distance between the two axes which permits making the gearing and its mounting in the form of the compact structure shown in Figs. 6 and 7. In this connection it is to be noted that the power capacity of the new gearing is far less dependent on the distance between the axes of the two parts than is the case in ordinary worm gearing. This is due to the fact that the extent of the tooth contact is dependent upon the lead of the worm rather than upon the distance between the axes, and it is a simple matter to select the proper lead in accordance with the equation above given after this distance has been chosen. Hence, when the space available for the gearing is limited by other features of the construction of a machine with which the gearing is to be used, the distance between the axes may be determined to fit the gearing into the required space, and the lead may then be made such as to give complete tooth contact and high power capacity.

An illustrative method for deriving Equation 2 will be described in connection with diagrammatic drawings,

Figs. 9 to 12, which illustrate a gearing in which the axes are perpendicular.

Pitch surfaces in the sense used herein are surfaces of revolution tangent to one another along a line. Thus, given a pair of skew axes located in space, it is possible to choose one pitch surface of arbitrary form, which will be referred to as the primary pitch surface, and determine the other from it mathematically. The contact line between such pitch surfaces may be straight or curved.

Referring .nowspecifically to Figs. 9 and 10, it will be seen that front and end views, respectively, of a pair of mating skew axis pitch surfaces are disclosed. The primary pitch surface is a cone, the axis of which is designated by the line 122. The axis of the conjugate pitch surface124is' designated by the numeral 126, ex-

tending at right angles to axis 122 at a distance designated by the letter C, Figs. 9 and 10. This distance C may be measured along a line 128, Figs. 9 and 10, such line being the common perpendicular to axes 122 and 126. The

dotted line represents the contact line between the pitch surfaces 120 and 124, and is the locus of points at which said surfaces are tangent to one another.

In the above mentioned diagrammatic representation, the vertex of thecone indicated by the numeral 132 is located an arbitrary distance a from the common perpe'n: dicular 128 and the half angle of the cone is an arbitrary angle designated by T.

Given the primary pitch surface 120 and the location of the two axes 122 and 126, the contact line 130 may be determined graphically or analytically. The graphic solution is shown in Figs. 9 and 10. Any point on the Contact line 130, such, for example, as the point 134 shown in the diagram, may be located by first choosing an arbitrary reference plane at a distance x from the common perpendicular 128, as shown in Fig. 9. Line 136 of Fig. 9 represents the aforesaid plane as seen on edge in the front view. At the point 138 where this plane intersects the periphery of the cone, a line 140 is drawn which is perpendicular-to the peripheral line 142 forming an element of the conical surface 120. The line 140 intersects the axis 122 at 144, and the radial line 146 extending from the point 144 to the axis 126 intersects the plane represented by the line 136 at the point 134. This point 134 is at a point in the reference plane 136, and the conical surface 120, at which the surface 120 is tangent to its conjugate surface 124.

By choosing other values of x and following the procedure outlined above, other contact points may be located in Fig. 9. In this manner a projection of the contact line or locus 130 may be plotted in the front view diagram of Fig. 9. In Fig. 10 the radius of the conical surface 120 in the plane determined by the line 136 is designated by the letter r. The point 134 may be located on the end view of the cone in Fig. 10 by merely projecting the point 134 in Fig. 9 until it intersects the arc of the radius r of Fig. 10. Thus two views of the contact line 130 are presented and these are sufficient to determine its entire shape and location. The surface 124 is the surface of revolution swept out by the line 130 rotated about the axis 126, and the surfaces 120 and 124 are tangent to one another along all points on the line 130.

Analytically, the location of the points on the locus or contact line 130 is attained by the following formulas:

Assuming: x, y and z are coordinates of a point as shown in Figs. 9 and 10. In this connection it will be noted that y is 'the distance of the point 134 from the axis 122, and z is the distance of the point 134 from a horizontal plane coincident with the axis 122.

T=the half-angle of the cone as shown in Fig. 9.

C=the distance between the axes 122 and 126.

a=the distance of the cone vertex 122 from a common perpendicular 128.

From the foregoing, the following equations define the locus 130:

The conjugate pitch surface 124 is determined graphically as follows:

An arc 148 passing through point 134 struck from the axis 126, Fig. 9, intersects the line 128 at the point 150. The point 156 may be projected into the end view, Fig. 10. It will thus be apparent that the point 150 is located on the axial profile of the 'pitch'surface 124. This procedure may be repeated for several points until the complete profile of the pitch surface 124 is determined. Obviously, the point 150 and the previously located contact point 134 lie in a common plane indicated by the line 152, Fig. 10. As previously indicated,

,hcxizontal plane face. The surface 9*and l0, 'isthe surface of a ,frustum of a conecorrethis plane 152 is spaced n-bm the axis 122 a distance Ldsignated.:;by y, and the point 134' is .spaced from the coincident with the axis 122 a distance designated as 1.

Assume r =radius of the surface 124 in the transverse cross-section of the plane 152.

t -(Cz) In Figs. 11 and 12 front and end views, respectively, disclose graphically the definition of what might be called an ideal tooth curve on the primary skew axis pitch sur- 120, as {previously indicated in Figs.

sponding to the arbitrary conicalcpitch surface. Surface 120 corresponds with a part of the conjugate pitch surface 12,4' fFigs. -9 and 10 and these two surfaces 120 and 124 are tangent to one another along the locus or contact line 130 previously explained. Gears of this type are usually designed for a fixed angular speed ratio. If a fixed speed ratio ,isimposed on-the mating pitch surfaces, it niakes' possible the determination of the direction of their ,relative m'otion at every point along the locus or contact line 130.

InfFig. 11 .the aforesaid relative motion is indicated vectorially. Arbitrarily selecting a point 154'.on the locus 130, :let it .be .assumedlhat the surface 129 is roitating'tat afixed angular speed in. 'This defines the direction and length of a vector 156, Fig. 11, which is the velocity of a point or particle on the surface 120 located instantaneously at the point 154. Since the ratio is fixed, the surface 124 must rotate at a fixed angular speed we. Thus, the direction and length of a vector 158, which is the velocity of a point or particle n the surface 124 located instantaneously on the point 154, are determined. Vector 160 indicates the difference of the vectors 156 and 158, and represents the direction and magnitude of relative velocity of the contacting points or particles at the point 154.

From the foregoing it will be obvious that a relative velocity vector corresponding to the vector 160 may be determined at every point along the locus 130, and that its direction in space depends only upon the angular speed ratio of the two pitch surfaces 120 and 124 and not upon the magnitudes of the angular speeds. It will also be obvious that relative velocity vectors such as the vector 160 will be tangent to both pitch surfaces at all points along the locus 130.

It can be shown mathematically that the direction of the relative velocity defines a curve 162 on the pitch surface 120. This curve can be thought of as being drawn upon and rotating with the surface 129. As the surface 120 rotates, the curve 162 intersects the locus 130 at a continuously moving point. The direction of velocities shown in Fig. 11 indicates that these points of intersection travel toward the small end of the cone. Thus, at the intersecting point indicated by the numeral 164, curve 162 is co-directional with the relative velocity vector indicated by the numeral 166. The velocity vectors at the point 164 are designated respectively by numerals 168 and 170. The fundamental property of the curve 162 is that at every point at which it intersects the locus or contact line 130, it is co-directional with the relative velocity vector. Obviously other congruent curves having the same property could be indicated as intersecting the locus 130 at points occupying other index positions on the surface 120. Curves, such as the curve 162, may be referred to as ideal tooth curves. in other words, the direction of the ideal tooth curve at every point along the locus 130 is the same as the direction of the relative velocity vector.

Fig. 14 is a developed view of the conical pitch surface 120 of Figs. 9 to 12, inclusive. When this surface development is Wrapped around the cone, points F and G will coincide. Hence, the curve 162 is actually a continuous curve of generally spiral form. If the develop- 10 ment illustrated in Fig. .114 is rotated counterclockwise fabout the -vertex..132 while theloc'us 13.0 remainsfixed, the intersection point .of :thc icurve 162 moves .continuously alongthe l0cus.;130 ;toward:the vertex. :Curv'e 5.162 may. be made to .intersectaevery :point on the. locusE'1-"30 within :the limits of the .development:byrrotating the cone about itsaxis. Anotherposition ofathe ideal tooth curve .is designated by the numeral 162a.and intersects thedocus 130 .at the .point 162b. These curves :at every .point 'of "intersection with the locus130 are co-directional with the relative velocity :vector.' The .dotted linelin Fig. -14 shows a curveapproximating the curve 162a.

Referring :toFig. 111 it will .be seen that :vectors 160 and 166 mustbe tangent to thegear too'th surfaces if conjugate :action.is'.to .:occur at :the points 154 and 164. "There .isionly one curve which willsatis'fythis condition at all points of the locus 130 and that curve is referredto hefein .as the ideal tooth curve. :.That-i's to say, :a gear tooth surface containing this curve is capable of .lconjugate' action at all points along wthellocus 13.0,. This curve will generate a mating curve i72 on the conjugate pitch surface 124, Fig. 'l'l, when the pitch surfaces. arid 124Iare rolled together at the proper speed ratio. If the traces of the gear teeth on the primary pitch surface differ from the above mentioned ideal tooth curve, the conjugate action, if it occurs at all, must occur at points away from the locus 130. This means that the fia'nks' of the gear .teeth will consist more and more of generated fillets and less of surfaces having true conjugate action.

Teeth based on the ideal tooth curve would make the best offset skew axis gears. However, such 'teem aret-not easy to produce. A method and apparatus for producing them are disclosed in my co-pending applications Serial Nos. 411,145 and 411,167, filed February 18, 1954. The present application is concerned with the method of producing a tapered helix of constant lead formed on a conical primary pitch surface. In development of this worm, the curve is in the form of an Archimedean spiral. Thus, the trace of the thread 21 of the tapered Worm will be co-directional with the ideal tooth curve at only one point, called the pitch point. This point will always lie on the locus as previously described. Its position on this locus can be arbitrarily chosen, but its selection is of importance in properly positioning the zone of action of the mating teeth. The lead of the tapered worm depends primarily upon two factors, namely, the required speed ratio and the location of the pitch point.

Making reference to Figs. 9 and 10, the formulas for determining the lead of the worm may be developed as follows:

:c, y, z=coordinates of and 10.

K: angular speed of pitch cone surface 120 angular speed of conjugate surface 124 r =1'adius of pitch cone in the transverse plane containing the pitch point.

L=the lead, measured along the axis of the cone, of a, tapered helix which is (to-directional with the ideal pitch point as defined in Figs. 9

stants a and T. The above equation for L of the specification.

The invention is hereby claimed as follows: 1. The method of making a face-type gear for use with a frusto-conical worm having a thread of uniform lead, which comprises hobbing a gear blank with a substitution of these values in the gives Equation 2 stated in column 8 11 frusto-conical hob of taper angle T having a thread of uniform lead L and of uniform cross-sectional shapedefined by a pressure angle between 20 and 45 on its high side and a pressure angle between and 25 on its low side, rotating the hob and the gear blank during the bobbing at a speed ratio K in excess of 9:1 and feeding the hob into a position in which its axis is at an angle E between 45 and 135 to the axis of the gear at a distance C therefrom which is less than the outer radius R of thesgear blank and in which its smaller and larger ends are atthe same side of the common perpendicular to the axes of the gear and worm and at distances x1 and x2 from this line and in which the apex of the outer conical surfaced the hob lies at the opposite side of this line at a distance a therefrom, the distances C, x1, x2 and a being related to the axis angle E, the speed ratio k and the taper angle T and lead L of the hob in accordance with the following equation:

21(C-P sin y K cs0 E--[% cos sin 1) cot E] where P=(a+a:) tan T 12 and x has a value between an and x2, so that the teeth formed on the gear are substantially free from inoperative fillets.

2. The method as claimed in claim 1 in which the axis of the hob is maintained at right angles to the axisof the gear blank and the distances C, x1, x2 and a are related to the speed ratio K and the taper angle T and lead L of the hob in accordance Wlth the following equation:

Where x has a value between x1 and am, so that the teeth formed on the gear are substantially free from inoperative fillets.

References Cited in the file of this patent a v ,1; Manama... 

